The supply and demand for bubbles (in pictures)

If the natural rate of interest, in an economy without a bubble, is below the growth rate of the economy, then the economy "needs" a bubble. There will be both a demand for bubbles, and a supply of bubbles, at any interest rate between that natural rate and the growth rate.

The vertical axis shows the real rate of interest. The horizontal axis shows the real value of the stock of assets.

Economists normally define desired savings as a flow demand for assets. Here I want to think of it as a stock demand for assets. Think of an overlapping generations model in which people desire to accumulate a certain value of a stock of assets for their retirement. And for emergencies and other lean years. I have drawn the savings curve as upward-sloping, so people's desired value of their stock of assets is an increasing function of the rate of interest. But that is not essential to my argument.

Economists also normally define desired investment as a flow. Here I want to think of it as a stock supply of real assets. It's the fundamental value of the stock of capital plus land, where "fundamental value" means the present value of the flow of earnings. The investment curve slopes down for two reasons: because a lower interest rate means a higher present value of a given flow of earnings; and because more investments become profitable at a lower rate of interest.

If all assets were priced at their fundamental values, so there are no bubbles, Ponzi schemes, chain letters, etc., then the equilibrium rate of interest would be the "no bubble" natural rate at the intersection of the savings and investment curves.

The supply curve of bubbles (Ponzi schemes, chain letters, etc.) is perfectly elastic at the growth rate of the economy. Here's why.

A bubble, Ponzi scheme, chain letter, etc., that grows at rate b will also pay an interest rate of b (ignoring transactions costs etc.). If it grows faster than the growth rate of the economy, it is unsustainable. Eventually the total value of the bubble will be bigger than the income of the next generation to buy into the bubble, so they won't be able to keep the bubble growing, and it will burst. But if the bubble grows more slowly than the economy, it can, in principle, continue indefinitely. It is profitable to start a bubble, Ponzi scheme, or chain letter. And if the rate of interest offered by the bubble is greater than the market rate of interest, people will willingly participate. So there will be an infinite supply of bubbles at any rate of interest below the growth rate of the economy.

The demand for bubbles, understood as the value of the stock of bubble assets that people will willingly hold, at the market rate of interest, is equal to the gap between the savings curve and the investment curve. For a given interest rate, a fall in desired investment, or an increase in desired savings, will increase the demand for bubbles.

Suppose initially there are no bubbles, and that all assets are priced at their fundamental values. The economy is at the intersection of the savings and investment curves. The equilibrium market rate of interest equals the no-bubble natural rate.

If the growth rate is below the no-bubble natural rate of interest, then it stays there. The market rate of interest is the no-bubble natural rate. The equilibrium quantity of bubbles is zero.

Now suppose the growth rate is above the no-bubble natural rate, as drawn above. A bubble asset that offers a rate of interest b, just slightly above the market rate, but below the growth rate, will be both supplied and demanded. As the bubble spreads and satisfies the demand for assets, the market rate of interest increases. As the market interest rate rises, the fundamental value of the stock of non-bubble assets falls. The value of household assets, including both fundamental plus bubble assets, may rise (if the Savings curve slopes up, as shown). But the flow of savings under a national accounts definition, which ignores capital gains on bubble assets, will fall and may become negative.

Competition between bubbles drives the market rate of interest up to the growth rate. At this point, all bubbles are on the margin of sustainability.

If something very small goes wrong, the bubbles will become unsustainable, and will burst. The interest rate immediately drops to the no-bubble natural rate. Then when confidence recovers it all starts again.

There's no money in the above model. No liquidity preference, only a loanable funds (plus bubble) theory of the rate of interest. With no medium of exchange, there's no distinction between aggregate demand and aggregate supply of newly-produced goods. So there can't be a recession caused by a deficiency of aggregate demand. That's the next step. But I think you can see that with money in the model it would be very hard for the rate of interest to fall quickly enough to prevent a recession, barring a very aggressive monetary policy.

Nick, could the natural rate of interest be below the growth rate? And if I understood the theory correctly, people will put money in a bubble that they could otherwise have put in a productive business? (My understanding was people invest in bubbles when there are no businesses with growth prospects).

Or perhaps the growth level in your graph represents a higher money supply level than natural?

Phillip: No. There is no money in that model, and no central bank. The bubble is trying to fix the problem of the interest rate being below the growth rate. It's a fundamental problem, not due to some policy mistake. Even a hypothetical central planner would need to fix it by creating a bubble.

Rogue: Yes, the equilibrium rate of interest could easily be below the growth rate of GDP, at least in an overlapping generations model. My simplified version of Samuelson's model in my previous bubble post had an equilibrium interest rate of minus 100%, and a growth rate of zero, for example. And in the 'real world' the real interest rate on safe government bonds is sometimes below the growth rate (or the nominal interest rate below the nominal growth rate). And people put their savings into bubbles because real fundamental investments can't satisfy all the desired savings at a rate of interest equal to (or just below) the growth rate.

I don't get it. How do you get economy growth rate way up high, relative to the rate of 'fundamental' return on capital stock. In other words - why your hypothetical economy so dynamically inefficient?

I think you need smth else - like perceptions of economy growth rate being disconnected from the real growth rate. Like a shock to a long-run rate of growth not being realized by citizens for a while.

"And if the rate of interest offered by the bubble is greater than the market rate of interest, people will willingly participate."

How about if the rate of return offered by the financial asset bubble is greater than the market rate of interest, people will willingly participate by becoming "bank like" using currency denominated debt?

By bank like I mean borrowing and trying to make "money" from a spread. That is borrow at 4% and hope to make 7% or more in the financial asset.

Dan: "In a two period OLG - easy. In a realistic multi-cohort model - not so easy."

Can you explain further? To my mind it's not obvious whether r would be less than or greater than g, with no bubbles, in a multi-cohort world. For example, suppose people save at a constant rate, then retire and dissave at a constant rate. If retirement lasts for (say) 20 years, the average person would hold 10 year's income as a stock of savings. If we assume r=g, would the stock of capital, plus houses, plus land, equal more or less than 10 years' GDP, evaluated at fundamental values? Then if we add in some extra savings for emergencies, whatever, it might be more than 10 years' GDP.

Plus, when we look at actual safe interest rates, they are not obviously greater than g.

I haven't made my mind up on this one. To my mind, r less than g is not so implausible we can rule it out.

I'm no expert on OLG, so I'm just re-producing smth I've heard in a seminar recently - if you put realistic demography + calibrated social security into OLG, you don't get sizable deviations from r=g. How valid is this - no idea.

On data - again I'm no expert on OLG and there might have been newer research on this - but I remember a paper by Abel, Mankiw and someone else in ReStud in early 90s (I think) and they said that there's no sign of dynamic inefficiency in the US and other developed countries.

If we consider the case were 'g' derives from the flow of 'i', then we can consider that if the flow of 'i' decreases (or becomes negative) that 'g' falls. Under these assumptions the natural-rate does not rise to meet g, g falls until the ponzi scheme cannot be sustained above the natural-rate.

This is a much more intuitive result. Consider that the original narrative suggested that the the ponzi scheme allows the elderly to increase their present consumption. From where does that present consumption come? It comes at the cost of diverting the young's investment. We KNOW that its a diversion of investment rather than present consumption because that's implicit in the natural-rate and the proliferation of the ponzi scheme itself. So we necessary see a shift along the production possibilities frontier in favor of present consumption and future relative poverty. Just what would be expected.

If g is below the natural rate, one could call the difference between the curves demand for consumption. Assets would tend to be liquidated leading to increased spending and growth. If g is above the natural rate, the demand for investment (bubbles) would tend to increase the production of capital assets supporting growth during the bubble until it popped which would tend to lower g to the natural rate.

Innovation would result in a shift in the supply of assets, i, to the right, obsolescence or a supply shock, a shift to the left. An ageing demographic would result in a shift in the demand for assets, s, to the right, a younger demographic, a shift to the left.

Jon: I was reminding my self to say that I couldn't really justify my assumption that g is exogenous, except to keep it simple. Then I forgot to say it!

What you say does indeed make a lot of intuitive sense. I think it's right. It's also a very interesting modification. One would have to distinguish between short run growth rates and steady state growth rates. That makes it a bit tricky to talk about the indefinite sustainability of bubbles. Bubbles (in aggregate) create the seeds of their own destruction, by bringing g down to no-bubble natural rate. The wage income of the young depends on the capital stock held by the old. I like it. But it's harder to think about.

Lord. But if g is below the no bubble natural rate, the equilibrium rate of interest will equal the natural rate. That's where savings equals investment. There won't be any excess demand for consumption.

And if g is above the natural rate, the bubbles created would reduce real investment. Those bubbles aren't productive. They would reduce output, as Jon says.

Fed: you have to distinguish real from nominal interest rates. My model above is talking about real interest rates. Central banks have no control over the real interest rate in the long run.

I was interpreting g as short run temporary growth which could fall below the natural rate during a recession and the natural rate as long run growth. So you are interpreting bubbles as purely price phenomena? I can see them both as price in some areas and in stock in others, high urban housing prices and large increases in housing stock in exurban areas, for example. The former changes prices but would not change investment much since for each sale there is a buyer and a seller and what one gains the other costs. The latter is real investment that would not necessarily be undertaken without a bubble. Once it collapses, it will reduce investment to a level lower than without one, but not until then. Bubbles and g (in the short run) would have positive feedback both on the way up and the way down.

Well, I have some issues with this and your previous post -- not meant to be nitpicky:

1. Definitions: instead referring to "chain-mail" swindles, just call them financial assets. We know what you mean, but precision helps here.

2. Definition: there is a whole term structure of rates, not one rate. So which rate are you talking about? IMO, "the" rate is the long term consol rate (e.g. t-->+infinity). I say this because the arbitrage functions that drive g to r (or that drive APK to r to g) only apply if r is the consol rate. The arguments break down when you need to start repaying principle, and the longer you can push principle repayment off, the better the arbitrage opportunity.

In that case, the mere existence of a sloping yield curve ( or equivalently, of diminishing returns for each particular investment) does not require any bubbles. Of course, as we do not sell consols, we proxy by other long term rates and just assume a few error bars in our results.

3. I disagree with your green investment curve sloping downward. A simple model with reproducible capital should assume that it is vertical, as this curve will only slope downward if capital is mispriced.

4. Definition of Bubble-assets: Instead of inventing chain-mail operations, let's just call it housing and be done with it.

5. Definition of "Investment" -- I am assuming that you mean "productive" investment -- i.e. cash-flow generating investment. In general, you are pointing out that a financial surplus need not be matched by a real (e.g. productive) investment. The surplus could exceed the amount invested productively, causing the remainder to be parked in a bubble asset (e.g. in a modern economy, this would be land).

6. Definition of "ponzi" -- The only sensible definition that I am aware of is that your operating cash-flow (after deducting interest expense) be positive. If you are reliant on issuing more debt (e.g. raising cash via financing) than you are in a ponzi state. Simply rolling over debt is not ponzi as all businesses and households do this, and there is nothing wrong with it.

This applies only to the private sector. For government, I'm not sure what it would mean to call it ponzi -- you can argue that the analogous result would require that tax inflows exceed interest outflows -- again, rolling over debt is not sufficient to launch the ponzi accusation.

7. I would argue differently from you -- that the natural *consol* rate is g, and on average will be g (over long periods).

8. You are arguing that bubbles form when interest rates are too high -- above the natural rate -- this is absurd, and in any case it is difficult for an economy to remain in a situation in which r(consol) != g due to arbitrage reaction function. I think I know why you are arguing this -- you want to assume that people save "too much" creating a demand for bubble assets, and that this can only happen if interest rates are too high. I disagree entirely with this line of reasoning, but that is because I model the capital markets as being perfectly elastic -- at least over medium time periods, and to a first approximation.

You *ought* to be able to capture the prominent dynamics of a bubble-dependent economy with a very simple model -- but it requires abandoning a lot of the Wicksellian goop, which is in any case inconsistent with elastic, arbitrage-free markets and endogenous capital. But I think you already know our disagreements in this area. Good post.

Nick's post said: "Fed: "What is the problem if low interest rates keep producing financial asset bubbles but raising them produces price deflation?" that would be exactly the problem."

I don't see any answer(s) there from anyone. If I go get some past comments/posts about apples, are they going to get deleted this time? I'd rather not waste time looking for them if they are going to be deleted. Thanks!

This is from U.S. Flow of Funds data, measuring household net-worth, as well as CPI-U and GS10, and realGDP.

On the y-axis is the 5 year simple moving average of the "real" GS10 rate -- a proxy for the consol rate -- minus the real GDP growth rate. "Real" was calculated by substracting out y/y CPI. This is crude, but the 5 year SMA helps even that out. I needed to subtract out the real GDP growth rate, because your diagram has it fixed, but in reality it has been changing, so view this is a normalization operation.

On the x-axis is the 5 year simple moving average of the growth rate (%) of real household net-worth, where here I could just deflate by CPI. I had to use the growth rate, and not the stock, otherwise it would shoot out of the graph -- as the economy grows, you expect the assets to grow as well. Ideally I would have calculated a rolling NPV sum discounted by realGDP, but this was good enough. Nevertheless, it sticks to your principle of measuring the market value of assets as opposed to the investment cash-flows.

There are marked swings to the left during the bubble expansion, and marked swings to the right during bubble contractions -- I added dates so that our favorite bubbles can be identified. This draws out the horizontal line you've drawn.

I also added a dotted vertical line representing the median asset growth rate over the period --3.5%, just slightly higher than the real GDP growth rate of 3.3% over the same period: 1952-2009. And another dotted line showing the median "real" GS10 rate = -0.7% over the period.

This should be a good pictoral demonstration of the horizontal expansions and contractions, but also raise the issue that the size of the bubble (e.g. the magnitude of the horizontal expansion) is not proportional to the distance between the real interest rate and the GDP growth rate (distance above or below zero in the picture).

More importantly, the risk-free overnight rate is MZM own rate -- it's what a investor can receive for parking money at zero maturity. The overnight lending rate between the central bank and the banks is a non-market, non-investor rate, that typically corresponds to somewhere between a 6 month to a 1 year commitment in the markets, but this is situation-specific -- when Volcker hiked the overnight rate, it was higher than all the investor yields, including the long bond. No investor could get that kind of return, regardless of their capital commitment.

Lord: Good point. If there is a bubble in a real asset (like houses) that may cause an increased production of houses.

RSJ: Glad you found these two posts!

1. But some financial assets can be priced at fundamental value. A chain letter is a financial asset with zero fundamental value. And a financial asset that does have a fundamental value, but is priced above that value, can be thought of as a fundamental value + a chain letter. (I just find it easier to think that way).

2. Agreed. The consol rate works best theoretically.

3. "A simple model with reproducible capital should assume that it is vertical, as this curve will only slope downward if capital is mispriced."

I think you meant to say "horizontal"? Assuming you did, I disagree. You only get a horizontal curve in a very simple case. Assume There is only one good, a plant called "shmoo", that grows at rate r. You can eat it, consume it, or let it grow. In that case the marginal product of capital, and the rate of return on investment, is a biological constant. I am making the standard assumption of diminishing returns, as you add more capital to a fixed amount of labour (or land), the MPK falls, so the value of the capital stock in equilibrium is a downward-sloping function of the rate of interest.

4. But housing was just the latest example. It might/will be something else next time.

I think the real difference here is in our aggregation techniques and models of production.

I'll try to be coherent, and let me know if you find anything objectionable.

The aggregation issues here are legion, but the most salient one here is:

In another post, you reminded Peter T that if MP > AP, then the firm would grow, and Peter T reminded you that growing firms are the norm, yet clearly no firm is infinitely large. The norm appears to be an "equilibrium" consisting of a mix of rapidly growing young firms young firms, more slowly growing firms, and mature firms that just pay out dividends.

Let's try to unravel the disconnect.

The "standard recipe" is to
A1. assume that a firm starts out at some initial state, grows to its optimal size via some process that no one cares about

A2. Once the firm achieves its optimal size it reaches an equilibrium state of timeless bliss.

A2. aggregate a collection of the bliss firms into an APF, or just assume that the whole economy is a single bliss firm.

A3. Introduce time only with an exogenous productivity constant in the APF (if even then).

A4. Do dynamic modeling with shocks, which are then interpreted as productivity shocks.

Here is another recipe:

1. Assume a *flow* of new firms being introduced, each of which tends towards its optimal firm size. The firm grows towards its optimal size via investment. This A1 process is exactly the macro investment process which has feedbacks into aggregate demand, etc. The sum of k(t) for each firm is the economy wide K(t). There is no exogenous productivity constant, as this is absorbed by the rate of introduction of new firms (e.g. the new firms contain the newly invented forms of capital/techniques).

2. For each firm, the equity value of the firm is the NPV of the terminal size -- i.e. the potential value. Only in this sense is the firm instantly at its optimal size (e.g. only in the sense of its market value, as the markets are forward-looking). But the actual capital stock will in general be lower. But this introduces expectations into the game from the start.

3. The fact that for each firm, the market price exceeds k(t) means that the firm has some market power. I.e. an investor cannot buy k(t). He can only buy the equity price, e(t) > k(t). The difference is that the particular organization has some monopoly component. It is the growth in k(t) that measures the contribution to agg. demand, whereas e(t) is the rate of interest for the economy as a whole (on average). Only for those firms that "have arrived" does k(t) = e(t), but these firms, by definition, will not re-invest any further (net of depreciation) and cannot contribute to AD. MPk(t) goes to r(t) *from above*.

In the beginning, the equity return is realized via a capital gain. When the firm has reached a saturation level -- its optimum size -- the returns are cash dividends.

4. Aggregate right away. K = sum[k(t)]. Your aggregate production function will look like an infinite sum of firms, each in a different state of growth, all but finitely many of which are "not yet born" (i.e. 0). Note that you are forced to include time here because you never had the chance to remove it.

Now, instead of relying on an exogenous productivity shock, the equity value (and therefore the speed with which the firms converge to their optimal size, as well as the contribution to agg. demand) is dependent on expectations as well as technological factors. It is not just a technological transformation function! Only in the perfect future expectations case can you reduce things to technology alone. As soon as there is a statistical error, you naturally get expectations shocks as well as technological shocks, and these propagate in the same way that the external factor in the neoclassical model does. Moreover, the micro-version of diminishing returns becomes the macro yield curve. Finally, there is no magical equity premium mystery to worry about. Last but not least, the return is not a function of time-preference, but the role of time preference is to decide whether an investor wants to buy an income stock or a growth stock -- e.g. dividends now vs. growth and more dividends in the future.

Now, if I can convince you to accept this model, then it wont be hard for me to convince you that the investment curve is horizontal. But first, let's see if you buy the model.

RSJ: I am about to head out for the long weekend, and your comment deserves more thought than I can currently get my head to apply.

I keep coming back to Ricardo's(?) argument for diminishing returns: if there were no diminishing returns (to labour, on a fixed stock of land), you could grow enough food to feed England in a window box. Same with capital. If there were no diminishing returns to capital, with a fixed stock of labour, you could just give one worker all the capital and he could produce the whole of GDP by himself.

I think you have an interesting analysis of the dynamics. But yours is, I think, a one-factor model. It's a dynamic version of my "shmoo" model. If you add in a second factor, like labour, or land, then you get diminishing returns at the aggregate level as you add more and more capital to a fixed stock of labour and land.

No, it's not a one factor model, but a different approach to modeling the factors of production and aggregating them -- I'll write more a bit later.

You are right in that you can view it as a schmoo model -- financially. With perfect foresight, expected returns will be the same (and constant), causing the I curve to be horizontal. I is measured financially -- e.g. market value of equity shares -- not in "real terms" -- e.g. number of tractors deployed. If you change your foresight assumptions so that the consol rate jumps around a mean, then I will jump around in time as well. In that case, the bubble expands and contracts as S moves around from left to right. The intersection of I and S will, over time, trace out loops similar to the panel I posted earlier.

In fact, it is easy to get a demand for bubbles in infinite horizon models. Check out "Money and Capital as Competing Media of Exchange" by Ricardo Lagos and Guillaume Rocheteau.

The fundamental condition needed is that the productivity of capital investment is poor. (Consider a standard OLG endowment model -- the productivity of capital is zero -- and a bubble asset can improve welfare).

By a "bubble," I mean an asset price that is higher than the underlying fundamental value. This is clearly the case for valued fiat money (a pure bubble). Replace the Lagos and Rocheteau storage technology with a Lucas coconut tree. Equity shares can circulate as a medium of exchange and they will exhibit a "liquidity premium" (they will be overvalued relative to fundamentals).

David: I'm in two minds about how to treat liquidity in this type of model. Part of me wants to agree with what you are saying. The other part of me wants to consider "liquidity services" on a par with the services of a consumer durable, like a car, painting, or fridge, and include those services as part of the flow of returns that create the fundamental value of the asset.